Multipliers, paramultipliers, and weak-strong uniqueness for the Navier-Stokes equations

Journal of Differential Equations

Volumn 226, Issue 2, 2006, Pages 373-428

  Germain, Pierre ^a  

^a CENTRE DE MATHÉMATIQUES LAURENT SCHWARTZ   (France)

Author keywords

Leray solutions; Multipliers; Navier Stokes equations; Paraproduct; Sobolev spaces; Weak strong uniqueness

Indexed keywords

EID: 33646507669     PISSN: 00220396     EISSN: 10902732     Source Type: Journal    
DOI: 10.1016/j.jde.2005.10.007     Document Type: Article

References (44)

1. Auscher P., Dubois S., and Tchamitchian P. On the stability of global solutions to Navier-Stokes equations in the space. J. Math. Pures Appl. 83 (2004) 673-697
2. N. Chinese Ann. Math. Ser. B 16 (1995) 407-412
3. Berselli L., and Manfrin R. On a theorem by Sohr for the Navier-Stokes equations. J. Evol. Equ. 4 (2004) 193-211
4. Bony J.-M. Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires. Ann. Sci. École Norm. Sup. 14 (1981) 209-246
5. Bourdaud G. Analyse fonctionnelle dans l'espace euclidien. Publ. Math. Univ. Paris VII 23 (1987)
6. 0, personal communication, 2004
7. Caffarelli L., Kohn R.-V., and Nirenberg L. Partial regularity of suitable weak solutions of the Navier-Stokes equations. Comm. Pure Appl. Math. XXXV (1982) 771-831
8. Cannone M. Ondelettes Paraproduits et Navier-Stokes (1995), Diderot, Paris
9. Chemin J.-Y. Théorèmes d'unicité pour le système de Navier-Stokes tridimensionnel. J. Anal. Math. 77 (1999) 22-50
10. Coifman R., Lions P.-L., Meyer Y., and Semmes S. Compensated compactness and Hardy spaces. J. Math. Pures Appl. 72 (1993) 247-286
11. David G., and Journé J.-L. A boundedness criterion for generalized Calderón-Zygmund operators. Ann. of Math. 120 (1984) 371-397
12. S. Dubois, Equations de Navier-Stokes dans l'espace : Espaces critiques et solutions d'énergie finie, Thèse de doctorat, Université de Picardie Jules Verne, 2002
13. Dubois S. Uniqueness for some Leray-Hopf solutions to the Navier-Stokes equations. J. Differential Equations 189 (2003) 99-147
14. 3, ∞-solutions of Navier-Stokes equations and backward uniqueness. Russian Math. Surveys 58 (2003) 211-250
15. p. Arch. Ration. Mech. Anal. 45 (1972) 222-240
16. Foias C. Une remarque sur l'unicité des solutions des équations de Navier-Stokes en dimension n. Bull. Soc. Math. France 89 (1961) 1-8
17. Fujita H., and Kato T. On the Navier-Stokes initial value problem. Arch. Ration. Mech. Anal. 16 (1964) 260-315
18. Gala S., and Lemarié-Rieusset P.-G. Multipliers between Sobolev spaces and fractional differentiation. Prépubl. Univ. d'Evry Val d'Essonne 203 (2005)
19. Gallagher I., and Planchon F. On global infinite energy solutions to the Navier-Stokes equations. Arch. Ration. Mech. Anal. 161 (2002) 307-337
20. Gallagher I., Iftimie D., and Planchon F. Comportement asymptotique et stabilité des solutions globales des équations de Navier-Stokes. Ann. Inst. Fourier (Grenoble) 53 (2003) 1387-1424
21. m with applications to weak solutions. Math. Z. 187 (1984) 471-480
22. Koch H., and Tataru D. Well-posedness for the Navier-Stokes equations. Adv. Math. 157 (2001) 22-35
23. Kozono H., and Taniuchi Y. Bilinear estimates in BMO and the Navier-Stokes equations. Math. Z. 235 (2000) 173-194
24. Kozono H., and Sohr H. Remark on uniqueness of weak solutions to the Navier-Stokes equations. Analysis (Munich) 16 (1996) 255-271
25. Lemarié-Rieusset P.-G. Recent Developments in the Navier-Stokes Problem (2003), Chapman and Hall
26. Leray J. Sur le mouvement d'un fluide visqueux remplissant l'espace. Acta Math. 63 (1934) 193-248
27. Maz'ya V.G., and Shaposhnikova T.O. Theory of Multipliers in Spaces of Differentiable Functions. Pitman Monographs and Studies in Math. vol. 23 (1985), Pitman, Boston/London
28. Maz'ya V.G., and Verbitsky I.E. Capacitary estimates for fractional integrals, with applications to partial differential equations and Sobolev multipliers. Ark. Mat. 33 (1995) 81-115
29. Maz'ya V.G., and Verbitsky I.E. The Schrödinger operator on the energy space: Boundedness and compactness criteria. Acta Math. 188 (2002) 263-302
30. V.G. Maz'ya, I.E. Verbitsky, The form boundedness criterion for the relativistic Schrödinger operator, preprint, 2004
31. Meyer M. Une classe d'espaces fonctionnels de type BMO. Application aux intégrales singulières. Ark. Mat. 27 (1989) 305-318
32. Meyer Y. Wavelets, Paraproducts and Navier-Stokes Equations. Current Developments in Math. (1999), MIT Press, Cambridge, MA
33. Miura H. Remark on uniqueness of mild solutions to the Navier-Stokes equations. J. Funct. Anal. 218 (2005) 110-129
34. S. Montgomery-Smith, Conditions implying regularity of the three-dimensional Navier-Stokes equations, preprint
35. Prodi G. Un teorema di unicitá per le equazioni di Navier-Stokes. Ann. Mat. Pura Appl. 48 (1959) 173-182
36. Ribaud F. A remark on the uniqueness problem for the weak solutions of Navier-Stokes equations. Ann. Fac. Sci. Toulouse XI (2002) 225-238
37. Seregin G., and Šverák V. On smoothness of suitable weak solutions to the Navier-Stokes equations. preprint, available at. http://www.pdmi.ras.ru/~seregin/Recent%20Publications/recentpubl.htm
38. Serrin J. The initial value problem for the Navier-Stokes equations. In: Langer R.E. (Ed). Nonlinear Problems (1963) 69-98
39. Sohr H. A regularity class for the Navier-Stokes equations in Lorentz spaces. J. Evol. Equ. 1 (2001) 441-467
40. Stein E.M. Harmonic Analysis (1993), Princeton Univ. Press, Princeton
41. Triebel H. Theory of Function Spaces II (1992), Birkhaüser, Basel
42. von Wahl W. The Equations of Navier-Stokes and Abstract Parabolic Equations (1985), Vieweg, Braunschweig
43. Youssfi A. Function Spaces Related to Singular Integral Operators. Function Spaces, Differential Operators and Nonlinear Analysis (1993), Teubner, Stuttgart/Leipzig
44. 1-BMO and bilinear operators. J. Math. Anal. Appl. 190 (1995) 780-794